How to retrieve data from a samsung s5 with a defected eMMc?

So my phone just suddenly died on me, and when I try to turn it on it only goes into a black screen where it says "samsung custom".
Sometimes I get it to work in safe mode but not long enough to download data from it because it freezes, turns itself off or goes back to the "samsung custom" screen.
I can however go into recovery mode, that works without a problem.
The people in the lab I went to say that the eMMc is ruined and they can replace it, But that means loosing all my data and files (unfortunately not everything is backed up).
Is there and way to retrieve the data by connecting to a computer using the recover mode? Any software that might help with that maybe?

If you use Odin to flash TWRP (if it is available for the device) adb will work by default in TWRP. My suggestion is try to get into download mode and flash TWRP, then once you boot TWRP you can grab a copy of minimal adb and fastboot. Unzip it to a folder then press shift and right click. Select open cmd window here. Then type adb pull /sdcard/ to pull the contents of your internal memory. If you need apps and app data also you can also do adb pull /data/ to get all of that. This is only if you can get into download mode with vol-, home, and power.